每週問題 April 11, 2016

利用畢氏定理判定兩個正交的複向量。

Let \mathcal{V} be a complex inner product space. Show that two vectors \mathbf{x} and \mathbf{y} in \mathcal{V} are orthogonal if and only if

\Vert \alpha\mathbf{x}+\beta\mathbf{y}\Vert^2=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2

for all pairs of scalars \alpha and \beta.

 
參考解答:

使用內積性質,

\begin{aligned} \Vert \alpha\mathbf{x}+\beta\mathbf{y}\Vert^2&=\left\langle \alpha\mathbf{x}+\beta\mathbf{y},\alpha\mathbf{x}+\beta\mathbf{y}\right\rangle\\ &=\left\langle \alpha\mathbf{x},\alpha\mathbf{x}\right\rangle+\left\langle \beta\mathbf{y},\beta\mathbf{y}\right\rangle+\left\langle \alpha\mathbf{x},\beta\mathbf{y}\right\rangle+\left\langle \beta\mathbf{y},\alpha\mathbf{x}\right\rangle\\ &=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2+\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle+\overline{\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle}\\ &=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2+2\hbox{Re}(\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle). \end{aligned}

對於所有的 \alpha\beta\Vert \alpha\mathbf{x}+\beta\mathbf{y}\Vert^2=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2 等價於 \left\langle \mathbf{x},\mathbf{y}\right\rangle=0

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